Interval error observer-based aircraft engine active fault tolerant control method

ABSTRACT

The present invention provides an interval error observer-based aircraft engine active fault tolerant control method, and belongs to the technical field of aircraft control. The method comprises: tracking the state and the output of a reference model of an aircraft engine through an error feedback controller; compensating a control system of the aircraft engine having a disturbance signal and actuator and sensor faults through a virtual sensor and a virtual actuator; observing an error between a system with fault of the aircraft engine and the reference model through an interval error observer, and feeding back the error to the error feedback controller; and finally, using a difference between the output of the reference model of the system with fault and the output of the virtual actuator as a control signal to realize active fault tolerant control of the aircraft engine.

TECHNICAL FIELD

The present invention relates to an interval error observer-basedaircraft engine active fault tolerant control method, belongs to thetechnical field of aircraft control, and particularly relates to anactive fault tolerant control method which is applied when aircraftengines have actuator and sensor fault under disturbances.

BACKGROUND

As the only power plant of an aircraft, an aircraft engine directlyaffects the safety, reliability and economy of the aircraft. Althoughreliable control system design can reduce the incidence of system fault,the actual system has complex structure and operates at high intensity;the factors that may cause the fault in the system are increasedgreatly; the types of the fault become increasingly diverse; and thefault of components is inevitable. As a driving component of theaircraft engine, the actuator is closely related to the state adjustmentof the system. The actuator has large workload and complicatedstructure, and easiest to fail. Once the actuator fails, the entiresystem is collapsed, which causes a serious impact. The sensor isresponsible for receiving and transmitting information of the aircraftengine system. The presence of sensor fault directly affects the safetyand the reliability of the system. Therefore, it is of greatsignificance to improve the fault-tolerant capacity of the system andensure the stability and performance criteria of the system after thefault. Traditional fault tolerant control methods also face newchallenges.

In general, fault tolerant control research methods are classified intotwo categories: passive fault tolerant control and active fault tolerantcontrol. The idea of passive fault tolerant control is to pre-design acontroller based on pre-judged possible faults, and passive faulttolerant control is a controller design method based on a robust controltechnology. When a fault occurs, the designed controller is called tokeep an entire closed-loop system insensitive to the fault, therebyachieving the stability of the system. However, as the system becomesmore and more complex, the types and number of the faults that may occurare increased. Therefore, the traditional passive fault tolerant controlhas great limitations, that is, all possible fault conditions need to beconsidered in advance, resulting in certain conservation of thecontroller. To reduce the conservation of the control system, activefault tolerant control that reconfigures the system becomes a researchhotspot. The idea of active fault tolerant control is to realize onlinefault compensation by readjusting the parameters of the controller orreconfiguring the system after the fault occurs. That is, when there isno fault, the system is operated normally; and once the fault occurs,the system automatically adjusts or reconfigures a control law. Theaircraft engine can be generally described as a linear-parameter-varying(LPV) system. Existing research results use a gain self-schedulingH-infinite optimization method when processing active fault tolerantcontrol of the LPV system having actuator and sensor faults. The methodreadjusts the controller parameters when the system has the fault,thereby increasing the complexity of the system design. In addition, thecontrol system of the aircraft engine is often interfered by noisesignals. The existing methods have no ideal solution for the activefault tolerant control of the sensors and actuator faults of theaircraft engine when processing the interference signals.

SUMMARY

The technical problem of the present invention is: when the aircraftengine have the actuator and sensor faults and the control system isaffected by noise signal interference, to solve the defects of theexisting control method, the present invention provides an intervalerror observer-based aircraft engine active fault tolerant controlmethod which can ensure that the aircraft engine can track a referencemodel without changing the structure and parameters of the controller.Namely, the reconfigured system has the same state and output as anoriginal fault-free system, realizes a desired control objective,enables the system to have the capability to eliminate the faultsautonomously, enhances the operating reliability of the aircraft engineand reduces maintenance cost of the aircraft engine.

The technical solution of the present invention is:

An interval error observer-based aircraft engine active fault tolerantcontrol method comprises the following steps:

step 1.1: establishing an affine parameter-dependent aircraft enginelinear-parameter-varying (LPV) model

{dot over (x)}_(p)(t)=[A ₀ +ΔA(θ)]x _(p)(t)+[B ₀ +ΔB(θ)]u _(p)(t)+d_(f)(t)

y _(p)(t)=C _(p) x _(p)(t)+v(t)   (1)

where R^(m) and R^(m n) respectively represent a m-dimensional realnumber column vector and a m-row n-column real matrix; state vectorsx_(p)=[Y_(nl) Y_(nh)]^(T) ∈ R^(n) ^(x) , Y_(nl) and Y_(nh) respectivelyrepresent variation of relative conversion speed of low pressure andhigh pressure rotors; n_(x) represents the dimension of a state variablex; n_(y) represents the dimension of an output vector y; n_(u)represents the dimension of control input u_(p); control inputu_(p)=U_(pf) ∈ R^(n) ^(u) is a fuel pressure step signal; output vectorsy_(p)=Y_(nh) ∈ R^(n) ^(y) , A₀ ∈ R^(n) ^(x) ^(×n) ^(x) , B₀ ∈ R^(n) ^(x)^(×n) ^(x) and C_(p) ∈ R^(n) ^(x) ^(×n) ^(x) are known system constantmatrices; d_(f)(t) is a disturbance variable; the relative conversionspeed n_(h) of the high pressure rotor of the aircraft engine is ascheduling parameter θ∈ R^(p); system variable matrices ΔA(θ) and ΔB(θ)satisfy −ΔA≤ΔA(θ)≤ΔA and −ΔB≤ΔB(θ)≤ΔB; ΔA ∈ R^(n) ^(x) ^(×n) ^(x) is anupper bound of ΔA(θ); ΔB ∈ R^(n) ^(x) ^(×n) ^(x) is an upper bound ofΔB(θ); ΔA≥0; ΔB≥0; a state variable initial value x_(p)(0) satisfies x₀≤x_(p)(0)≤x ₀; x ₀, x ₀ ∈ R^(n) ^(x) are respectively known upper boundand lower bound of the state variable initial value x_(p)(0); d, d ∈R^(n) ^(x) are known upper bound and lower bound of an unknowndisturbance d_(f)(t); sensor noise v(t) satisfies |v(t)|<V; V is a knownbound; V>0;

step 1.2: representing the reference model of the fault-free system ofthe aircraft engine as

{dot over (x)}_(pref)(t)=A ₀ x _(pref)(t)+B ₀ u _(pref)(t)

y _(pref)(t)=C _(p) x _(pref)(t)   (2)

where x_(pref) ∈ R^(n) ^(x) is a reference state vector of thefault-free system; u_(pref) ∈ R^(n) ^(u) is control input of thefault-free system; y_(pref) ∈ R^(n) ^(y) is a reference output vector;an error feedback controller of the fault-free system of the aircraftengine is designed according to the aircraft engine LPV modelestablished in the step 1.1;

step 1.2.1: defining an error e_(p)(t)=x_(pref)(t)−x_(p)(t) between theaffine parameter-dependent aircraft engine LPV model and the referencemodel of the fault-free system of the aircraft engine to obtain errorstate equations of the fault-free system:

ė_(p)(t)=[A ₀ +ΔA(θ)]e _(p)(t)+[B ₀ +ΔB(θ)]Δu _(cp)(t)−ΔA(θ)x_(pref)(t)−ΔB(θ)u _(pref)(t)−d _(f)(t)

ε_(cp)(t)=C _(p) e _(p)(t)=v(t)   (3)

where Δu_(cp)(t)=u_(pref)(t)−u_(p)(t) andε_(cp)(t)=y_(pref)(t)−y_(p)(t);

step 1.2.2: representing state equations of the upper bound ē_(p) andthe lower bound e _(p) of the error vector e_(p) as:

{dot over ( e )}_(p)(t)=[A ₀ −LC _(p) ]ē _(p)(t)+[B ₀ +ΔB]Δu _(cp)(t)+Lε_(cp)(t)+|L|V−d (t)+ΔA |x _(pref)(t)|+ϕ_(p)(t)

{dot over ( e )}_(p)(t)=[A ₀ −LC _(p) ]e _(p)(t)+[B ₀ −ΔB]Δu _(cp)(t)+Lε_(cp)(t)−|L|V−d (t)−ΔA |x _(pref)(t)|−ϕ_(p)(t)   (4)

where ē_(p), e _(p) ∈ R^(n) ^(x) are respectively the upper bound andthe lower bound of the error vector e_(P), i.e., e_(p)(t)≤e_(p)(t)≤ē_(p)(t); ϕ_(p)(t)=ΔA(ē_(p) ⁺(t)+e _(p) ⁻(t)), ē_(p)⁺=max {0, ē_(p)}, ē_(p) ⁻=ē_(p) ⁺−ē_(p), e _(p) ⁺=max{0, e _(p)}, e _(p)⁻=e _(p) ⁺−e _(p); L ∈ R^(n) ^(x) ^(×n) ^(y) is an error gain matrix ofthe fault-free system and satisfies A₀−LC_(p) ∈ M^(n) ^(x) ^(×n) ^(x) ;M^(n) ^(x) represents a set of n_(x)-dimensional Metzler matrix; |L|represents taking absolute values of all elements of the matrix L;

step 1.2.3: respectively setting e_(pa)=0.5(ē_(p)+e _(p)) ande_(pd)=ē_(p)−e _(p); rewriting the formula (4) as:

{dot over (e)}_(pd)(t)=[A ₀ −LC _(p) ]e _(pd)(t)+2 ΔBΔu_(cp)(t)+ϕ_(pd)(t)+δ_(pd)(t)

{dot over (e)}_(pa)(t)=[A ₀ −LC _(p) ]e _(pa)(t)+B ₀ Δu _(cp)(t)+LC _(p)e _(p)(t)+δ_(pa)(t)   (5)

where

ϕ_(pd)(t)=2ΔA ( e _(p) ⁺⁽ t)+ e _(p) ⁻(t))

δ_(pd)(t)=2|L|V−d (t)+ d (t)+2 ΔA|x _(pref)(t)|  (6)

δ_(pa)(t)=−Lv(t)−0.5( d (t)+ d (t))

step 1.2.4: representing the output of the error feedback controller as:

Δu _(cp)(t)=K _(a) e _(pa)(t)+K _(d) e _(pd)(t)   (7)

representing the gain matrix of the error feedback controller as K_(d),K_(a) ∈ R^(n) ^(x) ^(×n) ^(x) ; setting e_(x)(t)=e_(p)(t)−e_(pa)(t),−0.5e_(pd)(t)≤e_(x)(t)≤0.5e_(pd)(t), and then

{dot over (e)}_(pa)(t)=[A ₀ +B ₀ K _(a)]e _(pa)(t)+B ₀ K _(d) e_(pd)(t)+LC _(p) e _(x)(t)+δ_(pa)(t)   (8)

step 1.2.5: rewriting formulas (5) and (8) as:

$\begin{matrix}{{{\overset{.}{\xi}}_{p}(t)} = {{{G_{p}(t)}{\xi_{p}(t)}} + {\delta_{p}(t)}}} & (9) \\{{{{G_{p}(t)} = {\begin{bmatrix}{A_{0} - {LC}_{p}} & 0 \\{B_{0}K_{d}} & {A_{0} + {B_{0}K_{a}}}\end{bmatrix}\  + {A_{pd}(t)}}}{{where}\mspace{14mu} {\xi_{p}(t)}} = \left\lbrack {{e_{pd}(t)}^{T},{e_{pa}(t)}^{T}} \right\rbrack^{T}},{{\delta_{p}(t)} = {\left\lbrack {\left( {{\delta_{pd}(t)} + {2\overset{\_}{\Delta \; B}\Delta \; {u_{cp}(t)}}} \right)^{T},{\delta_{pa}(t)}^{T}} \right\rbrack^{T}\mspace{14mu} {and}\mspace{14mu} {then}}}} & (10) \\{\begin{bmatrix}\varphi_{pd} \\{LC_{p}e_{x}}\end{bmatrix} = {A_{pd}\begin{bmatrix}e_{pd} \\e_{pa}\end{bmatrix}}} & (11)\end{matrix}$

step 1.2.6: S^(m×m) representing an m-dimensional real symmetric squarematrix; setting a matrix E,F ∈ S^(2n) ^(x) ^(×2n) ^(x) ; E,F>0representing that each element in E,F is greater than 0; constant λ>0;and obtaining a matrix inequality:

G _(p) ^(T) E+EG _(p) +λE+F<0   (12)

namely, setting each element in G_(p) ^(T)E+EG_(p)+λE+F to be less than0; solving the matrix inequality (12) to obtain the gain matrices K_(d),K_(a) of the error feedback controller so as to obtain the errorfeedback controller from (7);

step 1.3: describing the aircraft engine LPV model having disturbanceand actuator and sensor faults as:

{dot over (x)}_(f)(t)=[A ₀ +ΔA(θ)]x _(f)(t)+B _(f)(γ(t))u _(f)(t)+d_(f)(t)

y _(f)(t)=C _(f)(ϕ(t))x _(f)(t)+v(t)   (13)

where x_(f) ∈ R^(n) ^(x) is a state vector of a system with fault; u_(f)∈ R^(n) ^(u) is the control input of the system with fault; y_(f) ∈R^(n) ^(y) is an output vector of the system with fault; B_(f)(γ(t)) andC_(f)(ϕ(t)) are respectively actuator and sensor faults, expressed as

B _(f)(γ(t))=[B ₀ +ΔB(θ)]diag(γ₁(t), . . . , γ_(n)(t))

C _(f)(ϕ(t))=C _(p)diag(ϕ₁(t), . . . , ϕ_(n)(t))   (14)

where 0≤γ_(i)(t)≤1 and 0≤ϕ_(j)(t)≤1 respectively represent the failuredegree of the i th actuator and the jth sensor; γ_(i)=1 and γ₁=0respectively represent health and complete failure of the i th actuator;ϕ_(j) is similar; diag(γ₁, γ₂, . . . , γ_(n)) represents a diagonalmatrix with diagonal elements γ₁, γ₂, . . . , γ_(n); diag(ϕ₁, ϕ₂, . . ., ϕ_(n)) is similar; setting γ(t) and ϕ(t) estimated values respectivelyas {circumflex over (γ)}(t) and {circumflex over (ϕ)}(t), and then

B _(f)(γ(t))=B _(f)({circumflex over (γ)}(t))+B _(f)(Δγ(Δγ(t))

C _(f)(ϕ(t))=C _(f)({circumflex over (ϕ)}(t))+C _(f)(Δϕ(t))   (15)

where Δγ(t)=γ(t)−{circumflex over (γ)}(t) and Δϕ(t)=ϕ(t)−{circumflexover (ϕ)}(t) are respectively errors of estimation of γ(t) and ϕ(t); avirtual actuator and a virtual sensor are respectively designedaccording to the actuator and sensor faults;

step 1.3.1: designing the virtual sensor as:

{dot over (x)}_(vs)(t)=A _(vs)(θ)x _(vs)(t)+B _(f)({circumflex over(γ)}(t))Δu(t)+Qy _(f)(t)

{circumflex over (y)}_(f)(t)=C _(vs) x _(vs)(t)+Py _(f)(t)   (16)

where

A _(vs)(θ)=A ₀ +ΔA(θ)−QC _(f)({circumflex over (ϕ)}(t))

C _(vs) =C _(p) −PC _(f)({circumflex over (ϕ)}(t))   (17)

where x_(vs) ∈ R^(n) ^(x) is a state variable of a virtual sensorsystem; Δu ∈ R^(n) ^(u) is a difference in control inputs of a faultmodel and a fault reference model; ŷ_(f) ∈ R^(n) ^(y) is an outputvector of the virtual sensor system; Q and P are respectively parametermatrices of the virtual sensor;

step 1.3.2: an LMI region S₁(ρ₁, q₁, r₁, θ₁) representing anintersection of a left half complex plane region with a bound of −ρ₁, acircular region with a radius of r₁ and a circle center of q₁ and a fanregion having an intersection angle θ₁ with a negative real axis;representing a state matrix A_(vs) of the virtual sensor as a polytopestructure; A_(vsj)=A₀+ΔA(θ_(j))−Q_(j)C_(f)({circumflex over (ϕ)}(t)),where θ_(j) represents the value of the jth vertex θ; A_(vsj) representsthe value of the state matrix A_(vs) of the virtual sensor of the jthvertex; a necessary and sufficient condition for eigenvalues of A_(vsj)to be in S₁(ρ₁, q₁, r₁, θ₁) is that there exists a symmetrical matrixX₁>0 so that the linear matrix inequalities (18)-(20) are established,thereby obtaining a parameter matrix Q_(j) of the virtual sensor of thecorresponding vertex;

$\begin{matrix}{{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} + {X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} + {2\rho_{1}X_{1}}} < 0} & (18) \\{\begin{bmatrix}{{- r_{1}}X_{1}} & {{q_{1}X_{1}} + {\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}\end{bmatrix}X_{1}}} \\{{q_{1}X_{1}} + {X_{1}\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}\end{bmatrix}}^{T}} & {{- r_{1}}X_{1}}\end{bmatrix} < 0} & (19) \\{\mspace{20mu} {\left( {\begin{matrix}{\sin \theta_{1}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} +} \\{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\cos \theta_{1}\begin{Bmatrix}{{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} -} \\{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}}\end{Bmatrix}}\end{matrix}\mspace{20mu} \begin{matrix}{\cos \; \theta_{1}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} -} \\{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\sin \; \theta_{1}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} +} \\{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T}\end{Bmatrix}}\end{matrix}} \right) < 0}} & (20)\end{matrix}$

selecting Q_(j) of a vertex corresponding to θ_(j) as a parameter matrixof the virtual sensor;

step 1.3.3: representing the parameter matrix P of the virtual sensoras:

P=C_(p)C_(f) ^(†)  (21)

where † represents pseudo-inversion of the matrix;

step 1.3.4: designing the virtual actuator as

{dot over (x)}_(va)(t)=A _(va) x _(va)(t)+B _(va) Δu _(c)(t)

Δu(t)=Mx _(va)(t)+NΔu _(c)(t)

y _(c)(t)={circumflex over (γ)}_(f)(t)+C _(p) x _(va)(t)   (22)

where

A _(va) =A ₀ +ΔA(θ)−B _(f)({circumflex over (γ)}(t))M

B _(va) =B ₀ +ΔB(θ)−B _(f)({circumflex over (γ)}(t))N   (23)

where x_(va) ∈ R^(n) ^(x) is a state variable of the virtual actuatorsystem; Δu_(c) ∈ R^(n) ^(u) is the output of the error feedbackcontroller; y_(c) ∈ R^(n) ^(y) is an output vector of the virtualactuator system; M and N are respectively parameter matrices of thevirtual actuator;

step 1.3.5: an LMI region S₂(ρ₂, q₂, r₂, θ₂) representing anintersection of a left half complex plane region with a bound of −ρ₂, acircular region with a radius of r₂ and a circle center of q₂ and a fanregion having an intersection angle θ₂ with a negative real axis;representing a state matrix A_(va) of the virtual actuator as a polytopestructure; A_(vaj)=A₀+ΔA(θ_(j))−B_(f)({circumflex over (γ)}(t))M_(j),where θ_(j) represents the value of the jth vertex θ; A_(vaj) representsthe value of the state matrix A_(va) of the virtual actuator of the jthvertex; a necessary and sufficient condition for eigenvalues of A_(vaj)to be in S₂(ρ₂, q₂, r₂, θ₂) is that there exists a symmetrical matrixX₂>0 so that the linear matrix inequalities (24)-(26) are established,thereby obtaining a parameter matrix M^(i) of the virtual actuator;

$\begin{matrix}{{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} + {X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} + {2\rho_{2}X_{2}}} < 0} & (24) \\{\begin{bmatrix}{{- r_{2}}X_{2}} & {{q_{2}X_{2}} + {\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}\end{bmatrix}X_{2}}} \\{{q_{2}X_{2}} + {X_{2}\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}\end{bmatrix}}^{T}} & {{- r_{2}}X_{2}}\end{bmatrix} < 0} & (25) \\{\mspace{20mu} {\left( {\begin{matrix}{\sin \theta_{2}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} +} \\{X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\cos \theta_{2}\begin{Bmatrix}{{X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} -} \\{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}}\end{Bmatrix}}\end{matrix}\mspace{20mu} \begin{matrix}{\cos \; \theta_{2}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} -} \\{X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\sin \; \theta_{2}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} +} \\{X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T}\end{Bmatrix}}\end{matrix}} \right) < 0}} & (26)\end{matrix}$

selecting M_(j) of a vertex corresponding to θ_(j) as a parameter matrixof the virtual actuator;

step 1.3.6: representing the parameter matrix N of the virtual actuatoras:

N=B_(f) ^(†)B_(p)   (27)

where † represents pseudo-inversion of the matrix;

step 1.4: designing an interval error observer according to the aircraftengine LPV model having disturbance and actuator and sensor faults andthe reference model of the system with fault;

step 1.4.1: representing the reference model of the aircraft enginesystem having disturbance and actuator and sensor faults as:

{dot over (x)}_(ref)(t)=A ₀ x _(ref)(t)+B _(f)({circumflex over(γ)}(t))u _(ref)(t)

y _(ref)(t)=C _(f)({circumflex over (ϕ)}(t))x _(ref)(t)   (28)

where X_(ref) ∈ R^(n) ^(x) is a reference state vector of the systemhaving disturbance and actuator and sensor faults; u_(ref) ∈ R^(n) ^(u)is control input of the system having disturbance and actuator andsensor faults; y_(ref) ∈ R^(n) ^(y) is a reference output vector of thesystem having disturbance and actuator and sensor faults;

step 1.4.2: defining an error e(t)=x_(ref)(t)−x_(f)(t) between theaircraft engine LPV model having disturbance and actuator and sensorfaults and the reference model of the aircraft engine to obtain errorstate equations of the system with fault of the aircraft engine based onthe LPV model:

{dot over (e)}(t)=[A ₀ +ΔA(θ)]e(t)+B _(f)({circumflex over (γ)})Δu(t)−B_(f)(Δγ)u _(f)(t)−ΔA(θ)x _(ref)(t)−d _(f)(t)

ε_(c)(t)=c _(f)(ϕ(t))e(t)−C _(f)(Δϕ)x _(ref)(t)−v(t)   (29)

where Δu(t)=u_(ref)(t)−u_(f)(t) and ε_(c)(t)=y_(ref)(t)−y_(f)(t);

step 1.4.3: representing state equations of an upper bound e and a lowerbound ē of the error e between the aircraft engine LPV model havingdisturbance and actuator and sensor faults and the reference model ofthe aircraft engine as:

{dot over ( e )}(t)=[A ₀ −LC _(f)(ϕ(t))] e (t)+[B ₀+ΔB ]Δu _(c)(t)+L[ε_(e)(t)+C _(p) x _(vu)+(C _(p) −PC _(f)(ϕ(t)))x _(vs) ]+|L|V−d (t)+ΔA |x_(ref)(t)|+ΔB |u _(ref)|+ϕ(t)

{dot over ( e )}(t)=[A ₀ −LC _(f)(ϕ(t))] e (t)+[B ₀−ΔB ]Δu _(c)(t)+L[ε_(c)(t)+C _(p) x _(va)+(C _(p) −PC _(f)(ϕ(t)))x _(vs) ]−|L|V−d (t)−ΔA|x_(ref)(t)|−ΔB |u _(ref)|−ϕ(t)   (30)

where ϕ(t)=ΔA(ē_(v) ⁺(t)+e _(v) ⁻(t)), e_(v) is a difference among theerror state variable of the system with fault of the aircraft enginebased on the LPV model, the state variable of the virtual actuator andthe state variable of the virtual sensor; the upper bound of e_(v) isē_(v)(t)=ē(t)−x_(va)(t)−x_(vs)(t); the lower bound of e_(v) is e_(v)(t)=e(t)−x_(va)(t)−x_(vs)(t); A₀−LC_(f) ∈ M^(n) ^(x) ^(n) ^(x) ;

step 1.4.4: setting e_(a)=0.5(ē+e), e_(d)=ē−e, and obtaining theinterval error observer from (30);

ė_(d)(t)=[A ₀ −LC _(f)(ϕ(t))]e _(d)(t)+2 ΔBΔu_(c)(t)+ϕ_(d)(t)+ϕ_(d)(t)+δ_(d)(t)

ė_(a)(t)=[A ₀ −LC _(f) ]e _(a)(t)+B ₀ K _(a) e _(a)(t)+B ₀ K _(d) e_(d)(t)+δ_(a)(t)+LC _(p) x _(va) +L(C _(p) −PC _(f))+LC _(f) e(t)   (31)

where

ϕ_(d)(t)=2ΔA ( e _(v) ⁺(t)+ e _(v) ⁻(t))

δ_(d)(t)=2|L|V−d (t)+ d (t)+2 ΔA|x _(ref)(t)|+2 ΔB|u _(ref)(t)|  (32)

δ_(a)(t)=−Lv(t)−0.5( d (t)+ d (t))

step 1.5: using the aircraft engine state variable x_(f)(t) of theaircraft engine LPV model having disturbance and actuator and sensorfaults, the output variable y_(f)(t), the reference model state variablex_(ref)(t) of the system with fault, the virtual actuator state variablex_(va)(t) and the virtual sensor state variable x_(vs)(t) as inputs ofthe interval error observer; using the interval error observer outpute_(a)(t), e_(d)(t) as the input of the error feedback controller; usingthe error feedback controller output Δu_(c)(t) as the input of thevirtual actuator; inputting the difference between the reference modeloutput u_(ref)(t) of the system with fault and the virtual actuatoroutput Δu(t) as a control signal into the system with fault of theaircraft engine, thereby realizing active fault tolerant control of theaircraft engine.

Compared with the existing technology, the interval error observer-basedaircraft engine active fault tolerant control method designed by thepresent invention has the advantages:

(1) In the active fault tolerant control of the LPV system havingactuator and sensor faults, the traditional gain self-schedulingH-infinite optimization method is often used. The method readjusts thecontroller parameters when the system has the fault, thereby increasingthe complexity of the system design. The active fault tolerant controlmethod proposed by the present invention can reconfigure the systemwhich simultaneously has actuator and sensor faults without redesigningthe controller.

(2) When the system has actuator faults and sensor faults, the methodproposed by the present invention can enable the reconfigured system tohave the same state and output as the original fault-free system.

(3) The method proposed by the present invention considers the problemthat often appears in the noise signal interference of the controlsystem in actual engineering, and improves the robustness of the controlsystem.

DESCRIPTION OF DRAWINGS

FIG. 1 is an overall structural diagram of a system.

FIG. 2(a) and FIG. 2(b) are respectively contrasts of trajectories ofH=0,Ma=0,n₂=94% aircraft engine LPV model states x_(p1)(t) and x_(p2)(t)and trajectories of fault-free reference model states x_(pref,1)(t) andx_(pref,2)(t).

FIG. 3 is a flow chart of an error feedback controller algorithm.

FIG. 4(a) and FIG. 4(b) are respectively the estimated curves of errorstates e_(p1)(t) and e_(p2)(t), upper bound states ē_(p1)(t) andē_(p2)(t) and lower bound states e _(p1)(t) and e _(p2)(t) ofH=0,Ma=0,n₂=94% aircraft engine fault-free system.

FIG. 5 is a varying curve of an actuator fault factor γ₁ and a sensorfault factor ϕ₁.

FIG. 6(a) and FIG. 6(b) are respectively the contrasts of trajectoriesof aircraft engine states x_(f1)(t) and x_(f2)(t) at H=0,Ma=0,n₂=94%under both disturbances and actuator and sensor faults, and trajectoriesof fault-free reference model states x_(pref,1)(t) and x_(pref,2)(t).

FIG. 7(a) and FIG. 7(b) are respectively the estimated curves ofaircraft engine error states e_(pf1)(t) and e_(pf2)(t), upper boundstates ē_(p1)(t) and ē_(p2)(t) and lower bound states e _(p1)(t) and e_(p2)(t) at H=0,Ma=0, n₂=94% under both disturbances and actuator andsensor faults.

FIG. 8 is a flow chart of a virtual sensor algorithm.

FIG. 9 is a flow chart of a virtual actuator algorithm.

FIG. 10 is a flow chart of an interval error observer algorithm.

FIG. 11(a) and FIG. 11(b) are respectively the contrasts of trajectoriesof aircraft engine states x₁(t) and x₂(t) at H=0,Ma=0,n₂=94% afteractive fault tolerant control and trajectories of fault reference modelstates x_(ref,1)(t) and x_(ref,2)(t).

FIG. 12(a) and FIG. 12(b) are respectively the estimated curves ofaircraft engine error states e₁(t) and e₂(t), and upper bound statesē₁(t) and ē₂(t) and lower bound states e ₁(t) and e ₂(t) of an errorobserver at H=0,Ma=0,n₂=94% after active fault tolerant control.

DETAILED DESCRIPTION

The embodiments of the present invention will be further described indetail below in combination with the drawings and the technicalsolution.

The overall structure of the present invention is shown in FIG. 1, andcomprises the following specific steps:

step 1.1: establishing an affine parameter-dependent aircraft engine LPVmodel; and taking relative conversion speed n₂ of a high pressure rotorof the aircraft engine as a variable parameter θ to normalizing thespeed n₂=88%, 89%, . . . , 100% , i.e., θ ∈ [−1,1], to obtain a model:

$\begin{matrix}{{{\overset{.}{x}}_{p} = {{\left\lbrack {A_{0} + {\Delta {A(\theta)}}} \right\rbrack {x_{p}(t)}} + {\left\lbrack {B_{0} + {\Delta {B(\theta)}}} \right\rbrack {u_{p}(t)}} + {d_{f}(t)}}}{y_{p} = {{C_{p}{x_{p}(t)}} + {v(t)}}}{where}} & (33) \\{{{A_{0} = \begin{bmatrix}{{- {2.6}}748} & {{- {0.6}}877} \\1.0704 & {{- {4.4}}672}\end{bmatrix}},{{\Delta \; {A(\theta)}} = \begin{bmatrix}{0.5199\theta} & {{- {2.4}}061\theta} \\{0.1049\theta} & {{- {0.8}}365\theta}\end{bmatrix}}}{{B_{0} = \begin{bmatrix}{{0.0}033} \\0.0012\end{bmatrix}},{{\Delta \; {B(\theta)}} = \begin{bmatrix}{{- {0.0}}004\theta} \\{{- {0.0}}001\theta}\end{bmatrix}}}{C_{p} = \begin{bmatrix}0 & 1\end{bmatrix}}} & (34)\end{matrix}$

the state variable initial value is x_(p)(0)=[0, 0]^(T); the upper boundand the lower bound of a disturbance variable d_(f)(t) are d, d ∈ R^(n)^(x) ;

${\overset{¯}{d} = {{- \underset{¯}{d}} = \begin{bmatrix}0.001 \\0.001\end{bmatrix}}};$

and the sensor noise bound is V=0.01. ΔA(θ) and ΔB(θ) have established−ΔA≥ΔA(θ)≤ΔA and −ΔB≤ΔB(θ)≤ΔB.

$\begin{matrix}{{\overset{\_}{\Delta \; A} = \begin{bmatrix}0.5199 & 2.4061 \\0.1049 & 0.8365\end{bmatrix}},{\overset{\_}{\Delta \; B} = \begin{bmatrix}0.0004 \\0.0001\end{bmatrix}}} & (35)\end{matrix}$

Step 1.2: representing the reference model of the fault-free system ofthe aircraft engine as

{dot over (x)}_(pref)(t)=A ₀ x _(pref)(t)+B ₀ u _(pref)(t)

y _(pref)(t)=C _(p) x _(pref)(t)   (36)

where the state vector of the reference model is a constant valuex_(pref)(t)=[4,2]^(T). At H=0, Ma=0 and n₂=94%, contrasts oftrajectories of aircraft engine LPV model states x_(p1)(t) and x_(p2)(t)and trajectories of fault-free reference model states x_(pref,1)(t) andx_(pref,2)(t) are shown in FIG. 2. An error feedback controller of afault-free system of the aircraft engine is designed, and an algorithmflow of the error feedback controller is shown in FIG. 3.

Step 1.2.1: defining an error e_(p)(t)=x_(pref)−x_(p) between the affineparameter-dependent aircraft engine LPV model and the reference model ofthe fault-free system of the aircraft engine, with an initial value ise_(p)(0)=x_(pref)(0)−x_(p)(0)=[4, 2]^(T).

Step 1.2.2: representing state equations of the upper bound ē_(p) andthe lower bound e _(p) of the error vector e_(p) as:

{dot over ( e )}_(p)(t)=[A ₀ −LC _(p)] e (t)+[B ₀ +ΔB]Δu _(cp)(t)+Lε_(cp)(t)+|L|V− d (t)+ΔA |x _(pref)(t)|+ϕ_(p)(t)

{dot over ( e )}_(p)(t)=[A ₀ −LC _(p)] e (t)+[B ₀ −ΔB ]Δu _(cp)(t)+Lε_(cp)(t)−|L|V− d (t)−ΔA |x _(pref)(t)|−ϕ_(p)(t)   (37)

where e _(p)(0)=[−50, −50]^(T), ē_(p)(0)=[50, 50]^(T) andϕ_(p)(t)=ΔA(ē_(p) ⁺(t)+e _(p)(t)). At H=0, Ma=0 and n₂=94%, theestimated curves of error states e_(p1)(t) and e_(p2)(t), upper boundē_(p1)(t) and ē_(p2)(t) and lower bound e _(p1)(t) and e _(p2)(t) ofaircraft engine fault-free system are shown in FIG. 4. From A₀−LC_(p) ∈M^(n) ^(x) ^(×n) ^(x) , the error gain matrix of the fault-free systemcan be obtained

$\begin{matrix}{L = \begin{bmatrix}{- 5} \\{20}\end{bmatrix}} & (38)\end{matrix}$

Step 1.2.3: respectively setting e_(pa)=0.5(ē_(p)+e _(p)),e_(pd)=ē_(p)−e _(p) to obtain

{dot over (e)}_(pd)(t)=[A ₀ −LC _(p) ]e _(pd)(t)+2 ΔBΔu_(cp)(t)+ϕ_(pd)(t)+δ_(pd)(t)

{dot over (e)}_(pa)(t)=[A ₀ −LC _(p) ]e _(pa)(t)+B ₀ K _(a) e _(pa)(t)+B₀ K _(d) e _(pd)(t)+LC _(p) e _(p)(t)+δ_(pa)(t)

ϕ_(pd)(t)=2 ΔA ( e _(p) ⁺(t)+ e _(p) ⁻(t))   (39)

δ_(pd)(t)=2|L|V−d (t)+ d (t)+2 ΔA|x _(pref)(t)|

δ_(pa)(t)=−Lv(t)−0.5( d (t)+ d (t))

where ē_(p1) and e _(p1) (ē_(p2) and e _(p2)) respectively represent thefirst (second) element of ē_(p) and e _(p), and e_(x,1) and e_(x,2)respectively represent the first element and the second element ofe_(x).

$\begin{matrix}{{2\left( {{\overset{¯}{e}}_{p\; 1}^{+} + {\underset{\_}{e}}_{p\; 1}^{-}} \right)} = \left\{ {{\begin{matrix}{{2{\overset{¯}{e}}_{p1}} = {e_{{pd},1} + {2e_{{pa},1}}}} & {{{\overset{¯}{e}}_{p1} \geq 0},{{\underset{\_}{e}}_{p\; 1} \geq 0}} \\{{2\left( {{\overset{¯}{e}}_{p1} - {\underset{\_}{e}}_{p\; 1}} \right)} = {2e_{{pd},1}}} & {{{\overset{¯}{e}}_{p1} \geq 0},{{\underset{\_}{e}}_{p\; 1} < 0}} \\{{{- 2}{\underset{\_}{e}}_{p\; 1}} = {e_{{pd},1} - {2e_{{pa},1}}}} & {{{\overset{¯}{e}}_{p1} < 0},{{\underset{\_}{e}}_{p\; 1} < 0}}\end{matrix}2\left( {{\overset{¯}{e}}_{p\; 2}^{+} + {\underset{\_}{e}}_{p\; 2}^{-}} \right)} = \left\{ \begin{matrix}{{2{\overset{¯}{e}}_{p2}} = {e_{{pd},2} + {2e_{{pa},2}}}} & {{{\overset{¯}{e}}_{p\; 2} \geq 0},{{\underset{\_}{e}}_{p\; 2} \geq 0}} \\{{2\left( {{\overset{¯}{e}}_{p\; 2} - {\underset{\_}{e}}_{p\; 2}} \right)} = {2e_{{pd},2}}} & {{{\overset{¯}{e}}_{p\; 2} \geq 0},{{\underset{\_}{e}}_{p\; 2} < 0}} \\{{{- 2}{\underset{\_}{e}}_{p\; 1}} = {e_{{pd},2} - {2e_{{pa},2}}}} & {{{\overset{¯}{e}}_{p\; 2} < 0},{{\underset{\_}{e}}_{p\; 2} < 0}}\end{matrix} \right.} \right.} & (40)\end{matrix}$

Step 1.2.4: representing the output of the error feedback controller as

Δu _(cp)(t)=K _(a) e _(pa)(t)+K _(d) e _(pd)(t)   (41)

representing the gain matrix of the error feedback controller as K_(d),K_(a) ∈ R^(n) ^(x) ^(×n) ^(x) ; setting e_(x)(t)=e_(p)(t)−e_(pa)(t),−0.5e_(pd)(t)≤e_(x)(t)≤0.5e_(pd)(t), and then

{dot over (e)}_(pa)(t)=[A ₀ +B ₀ K _(a)]e _(pa)(t)+B ₀ K _(d) e_(pd)(t)+LC _(p) e _(x)(t)+δ_(pa)(t)   (42)

Step 1.2.5: rewriting (39) and (42) as

$\begin{matrix}{{{\overset{.}{\xi}}_{p}(t)} = {{{G_{p}(t)}{\xi_{p}(t)}} + {\delta_{p}(t)}}} & (42) \\{{{{G_{p}(t)} = {\begin{bmatrix}{A_{0} - {LC}_{p}} & 0 \\{B_{0}K_{d}} & {A_{0} + {B_{0}K_{a}}}\end{bmatrix}\  + {A_{pd}(t)}}}{{where}\mspace{14mu} {\xi_{p}(t)}} = \left\lbrack {{e_{pd}(t)}^{T},{e_{pa}(t)}^{T}} \right\rbrack^{T}}{{{\delta_{p}(t)} = \left\lbrack {\left( {{\delta_{pd}(t)} + {2\overset{\_}{\Delta \; B}\Delta \; {u_{cp}(t)}}} \right)^{T},{\delta_{pa}(t)}^{T}} \right\rbrack^{T}},{{{and}\mspace{14mu} {{then}\begin{bmatrix}\varphi_{pd} \\{LC_{p}e_{x}}\end{bmatrix}}} = {A_{pd}\begin{bmatrix}e_{pd} \\e_{pa}\end{bmatrix}}}}{{{2\overset{\_}{\Delta \; A}\left( {{{\overset{¯}{e}}_{p}^{+}(t)} + {{\underset{\_}{e}}_{p}^{-}(t)}} \right)} = {A_{pd1}\begin{bmatrix}e_{pd} \\e_{po}\end{bmatrix}}},{{{LC}_{p}e_{x}} = {A_{pd2}\begin{bmatrix}e_{pd} \\e_{pa}\end{bmatrix}}}}} & (44) \\{{{A_{pd1} = {\overset{\_}{\Delta \; A}\begin{bmatrix}a_{11} & 0 & a_{13} & 0 \\0 & a_{22} & 0 & a_{24}\end{bmatrix}}},{A_{pd2} = \begin{bmatrix}0 & a_{31} & 0 & 0 \\0 & a_{41} & 0 & 0\end{bmatrix}}}{A_{pd} = \begin{bmatrix}A_{pd1} \\A_{pd2}\end{bmatrix}}} & (45)\end{matrix}$

All possible combining forms are considered: (a_(ll), a₁₃) ∈ {(1, 2),(2, 0), (1, −2)}, (a₂₂, a₂₄) ∈ {(1,2),(2,0),(1,−2)} and (a₃₁, a₄₁) ∈{(−2.5,10),(2.5,−10)}.

Step 1.2.6: S^(m×m) representing an m-dimensional real symmetric squarematrix; setting a matrix E,F ∈ S^(2n) ^(x) ^(×2n) ^(x) ; E,F>0representing that each element in E,F is greater than 0; constant λ>0;and obtaining a matrix inequality:

G _(p) ^(T) E+EG _(p) +λE+F<0   (46)

namely, setting each element in G_(p) ^(T)E+EG_(p)+λE+F to be less than0; converting the matrix inequality (46) to a linear matrix inequality(LMI), and multiplying the left and right sides of the inequality (46)by E⁻¹ to obtain

$\begin{matrix}{{{E^{1}G_{p}^{T}} + {G_{p}E^{1}} + {\lambda E^{1}} + F_{p}} \prec 0} & (47) \\{{{G_{p}(t)} = {\begin{bmatrix}{A_{0} - {LC_{p}}} & 0 \\0 & A_{0}\end{bmatrix} + {A_{pd}(t)} + {\begin{bmatrix}0 \\B_{0}\end{bmatrix}K}}}{K = \begin{bmatrix}{K_{d}\ } & K_{a}\end{bmatrix}}} & (48)\end{matrix}$

introducing W=KE⁻¹, and then converting inequality (46) into the LMI;using an LMI tool kit to obtain

K _(d)=[−0.0014 −0.0002]

K _(a)=[−14.5130 −21.8837]

Step 1.3: describing the aircraft engine LPV model having disturbanceand actuator and sensor faults as:

{dot over (x)}_(f)(t)=[A ₀ +ΔA(θ)]x _(f)(t)+B _(f)(γ(t)u _(f)(t)+d_(f)(t)

y _(f)(t)=C _(f)(ϕ(t))x _(f)(t)+v(t)

B _(f)(γ(t)=[B ₀ +ΔB(θ)]diag(γ₁(t), . . . , γ_(n)(t))

C _(f)(ϕ(t))=C _(p)diag(ϕ₁(t), . . . , ϕ_(n)(t))   (50)

where state variable initial values x_(f)(0)=[0,0]^(T), B_(f)(γ(t)) andC_(f)(ϕ(t)) are respectively actuator and sensor faults; and theactuator fault factor γ₁ and the sensor fault factor ϕ₁ decay from 1 to0.2 in the 5th to 6th seconds, as shown in FIG. 5. At H=0, Ma=0 andn₂=94%, contrasts of trajectories of states x_(f1)(t) and x_(f2)(t) ofthe aircraft engine having disturbance and actuator and sensor faultsand trajectories of fault-free reference model states x_(pref,1)(t) andx_(pref,2)(t) are shown in FIG. 6. Estimated curves of error statese_(pf1)(t) and e_(pf2)(t) and upper bounds ē_(p1)(t) and ē_(p2)(t) andlower bounds e _(p1)(t) and e _(p2)(t) of the aircraft engine havingdisturbance and actuator and sensor faults are shown in FIG. 7. Avirtual sensor and a virtual actuator are respectively designedaccording to the actuator and sensor faults, and algorithm flows arerespectively shown in FIG. 8 and FIG. 9.

Step 1.3.1: designing the virtual sensor as

{dot over (x)}_(vs)(t)=A _(vs)(θ)x _(vs)(t)+B _(f)({circumflex over(γ)}(t))Δu(t)+Qy _(f)(t)

{circumflex over (y)}_(f)(t)=C _(vs) x _(vs)(t)+Py _(f)(t)   (51)

where

A _(vs)(θ)=A ₀ +ΔA(θ)−QC _(f)({circumflex over (ϕ)}(t))

C _(vs) =C _(p) −PC _(f)({circumflex over (ϕ)}(t))   (52)

where x_(vs) ∈ R^(n) ^(x) is a state variable of a virtual sensorsystem; Δu ∈ R^(n) ^(u) is a difference in inputs of a fault model and afault reference model; ŷ_(f) ∈ R^(n) ^(y) is an output vector of thevirtual sensor system; Q and P are respectively parameter matrices ofthe virtual sensor.

Step 1.3.2: selecting an LMI region S₁(10, −4.5,15, π/6) and solvingLMIs (53)-(55)

$\begin{matrix}{{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} + {X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} + {2\rho_{1}X_{1}}} < 0} & (53) \\{\begin{bmatrix}{{- r_{1}}X_{1}} & {{q_{1}X_{1}} + {\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}\end{bmatrix}X_{1}}} \\{{q_{1}X_{1}} + {X_{1}\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}\end{bmatrix}}^{T}} & {{- r_{1}}X_{1}}\end{bmatrix} < 0} & (54) \\{\mspace{20mu} {\left( {\begin{matrix}{\sin \theta_{1}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} +} \\{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\cos \theta_{1}\begin{Bmatrix}{{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} -} \\{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}}\end{Bmatrix}}\end{matrix}\mspace{20mu} \begin{matrix}{\cos \; \theta_{1}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} -} \\{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\sin \; \theta_{1}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} +} \\{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T}\end{Bmatrix}}\end{matrix}} \right) < 0}} & (55)\end{matrix}$

obtaining a parameter matrix of a virtual sensor of a correspondingvertex

Q ₁=[−15.4224; 24.4935]

Q ₂=[8.5894; 33.1359]  (56)

Step 1.3.3: representing the parameter matrix P of the virtual sensor as

P=C_(p)C_(f) ^(†)=5   (57)

where † represents pseudo-inversion of the matrix.

step 1.3.4: designing the virtual actuator as

{dot over (x)}_(va)(t)=A_(va) x _(va)(t)+B _(va) Δu _(c)(t)

Δu(t)=Mx _(va)(t)+N Δu _(c)(t)   (58)

y _(c)(t)={circumflex over (y)}_(f)(t)+C _(p) x _(va)(t)

where

A _(va) =A ₀ +ΔA(θ)−B _(f)({circumflex over (γ)}(t))M

B _(va) =B ₀ +ΔB(θ)−B _(f)({circumflex over (γ)}(t))N   (59)

where x_(va) ∈ R^(n) ^(x) is a state variable of the virtual actuatorsystem; Δu_(c) ∈ R^(n) ^(u) is the output of the error feedbackcontroller; y_(c) ∈ R^(n) ^(y) is an output vector of the virtualactuator system; M and N are respectively parameter matrices of thevirtual actuator.

Step 1.3.5: selecting an LMI region S₂(1.5,−2,8, π/6) and solving LMIs(60)-(62)

$\begin{matrix}{{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} + {X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} + {2\rho_{2}X_{2}}} < 0} & (60) \\{\begin{bmatrix}{{- r_{2}}X_{2}} & {{q_{2}X_{2}} + {\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}\end{bmatrix}X_{2}}} \\{{q_{2}X_{2}} + {X_{2}\begin{bmatrix}{A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\{{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}\end{bmatrix}}^{T}} & {{- r_{2}}X_{2}}\end{bmatrix} < 0} & (61) \\{\mspace{20mu} {\left( {\begin{matrix}{\sin \theta_{2}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} +} \\{X_{2}\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\cos \theta_{2}\begin{Bmatrix}{{X_{2}\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} -} \\{\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}}\end{Bmatrix}}\end{matrix}\mspace{20mu} \begin{matrix}{\cos \; \theta_{2}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} -} \\{X_{2}\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T}\end{Bmatrix}} \\{\sin \; \theta_{2}\begin{Bmatrix}{{\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} +} \\{X_{2}\left\lbrack {A_{0} + {\Delta \; {A\left( {\rho (t)} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T}\end{Bmatrix}}\end{matrix}} \right) < 0}} & (62)\end{matrix}$

obtaining a parameter matrix of a virtual actuator of a correspondingvertex

M ₁=[3690.6 −4333.2]

M₂=[2170.5 2186.6]  (63)

Step 1.3.6: representing the matrix N of the virtual actuator as

N=B _(f) ^(†) B _(p)=5   (64)

Step 1.4: designing an interval error observer, wherein an algorithmflow of the interval error observer is shown in FIG. 10.

Step 1.4.1: representing the reference model of the aircraft enginesystem having disturbance and actuator and sensor faults as

{dot over (x)}_(ref)(t)=A ₀ x _(f)(t)+B _(f)({circumflex over (γ)}(t))u_(ref)(t)

y _(ref)(t)=C _(f)({circumflex over (ϕ)}(t))x _(ref)(t)   (65)

where the state vector of the reference model is a constant valuex_(ref)(t)=[4,2]^(T).

Step 1.4.2: defining an error e(t)=x_(ref)−x_(f) between the aircraftengine LPV model having disturbance and actuator and sensor faults andthe reference model of the aircraft engine with an initial value of theerror e(0)=x_(ref)(0)−x_(f)(0)=[4, 2]^(T). At H=0, Ma=0 and n₂=94%, thecontrasts of trajectories of aircraft engine states x₁(t) and x₂(t) andtrajectories of fault reference model states x_(ref1)(t) and x_(ref2)(t) after active fault tolerant control are shown in FIG. 11.

Step 1.4.3: representing state equations of an upper bound ē and a lowerbound e of the error e between the aircraft engine LPV model havingdisturbance and actuator and sensor faults and the reference model ofthe aircraft engine as

{dot over ( e )}(t)=[A ₀ −LC _(f)(ϕ(t))] e (t)+[B ₀ +ΔB]Δu _(c)(t)+L[ε_(c)(t)+C _(p) x _(va)+(C _(p) −PC _(f)(ϕ(t)))x _(vs) ]+|L|V−d (t)+ΔA |x_(ref)(t)|+ΔB |u _(ref)+ϕ(t)   (66)

{dot over ( e )}(t)=[A ₀ −LC _(f)(ϕ(t))] e (t)+[B ₀ −ΔB]Δu_(c)(t)+L[ε_(c)(t)+C_(p) x _(va)+(C _(p) −PC _(f)(ϕ(t)))x _(vs) ]−|L|V−d(t)−ΔA |x _(ref)(t)|−ΔB |u _(ref)|−ϕ(t)

where ϕ(t)=ΔA(ē_(v) ⁻(t)+e _(v) ⁻(t)),ē_(v)(t)=ē(t)−x_(va)(t)−x_(vs)(t), e _(v)(0)=[−50, −50]^(T) andē_(v)(0)=[50,50]^(T). e _(v)(t)=e(t)−x_(va)(t)−x_(vs)(t). The gainmatrix L of the observer satisfies A₀−LC_(p) ∈ M^(n) ^(x) ^(×n) ^(x) .

Step 1.4.4: setting e_(a)=0.5(ē+e), e_(d)=ē−e, and obtaining theinterval error observer from (66)

{dot over (e)}_(d)(t)=[A ₀ −LC _(f)(ϕ(t))]e _(d)(t)+2 ΔBΔu_(c)(t)+ϕ_(d)(t)+δ_(d)(t)

{dot over (e)}_(a)(t)=[A ₀ −LC _(f) ]e _(a)(t)+B ₀ K _(a) e _(a)(t)+B ₀K _(d) e _(d)(t)+δ_(a)(t)+LC _(p) x _(va) +L(C _(p) −PC _(f))+LC _(f)e(t)   (67)

where

ϕ_(d)(t)=2 ΔA ( e _(v) ⁺(t)+ e _(v) ⁻(t))

δ_(d)(t)=2|L|V−d (t)+ d (t)+2 ΔA|x _(pref)(t)|+2 ΔB|u _(pref)(t)|  (68)

δ_(a)(t)=−Lv(t)−0.5( d (t)+ d (t))

At H=0, Ma=0 and n₂=94%, estimated curves of aircraft engine errorstates e₁(t) and e₂(t), and upper bound states ē₁(t) and ē₂(t) and lowerbound states e ₁(t) and e ₂(t) of an error observer after active faulttolerant control are shown in FIG. 12.

Step 1.5: showing the overall structure that realizes the active faulttolerant control of the aircraft engine in FIG. 1.

Simulation results show that when the actuator and the sensor of theaircraft engine fail, an overshooting process occurs in states andoutputs after active fault tolerant control, but the actuator and thesensor quickly return to a normal state. This indicates that theinterval error observer-based aircraft engine active fault tolerantcontrol method can ensure that the reconfigured system has the sameperformance criteria as the original fault-free system.

1. An interval error observer-based aircraft engine active fault tolerant control method, comprising the following steps: step 1.1: establishing an affine parameter-dependent aircraft engine linear-parameter-varying LPV model {dot over (x)}(t)=[A ₀ +ΔA(θ)]x _(p)(t)+[B ₀ +ΔB(θ)]u _(p)(t)+d _(f)(t) y _(p)(t)=C _(p) x _(p)(t)+v(t)   (1) where R^(m) and R^(m×n) respectively represent a m-dimensional real number column vector and a m-row n-column real matrix; state vectors x_(p)=[Y_(nl) Y_(nh)]^(T) ∈ R^(n) ^(x) , Y_(nl) and Y_(nh) respectively represent variation of relative conversion speed of low pressure and high pressure rotors; n_(x) represents the dimension of a state variable x; n_(y) represents the dimension of an output vector y; n_(u) represents the dimension of control input u_(p); control input u_(p)=U_(pf) ∈ R^(n) ^(u) is a fuel pressure step signal; output vectors y_(p)=Y_(nh) ∈ R^(n) ^(y) , A₀ ∈ R^(n) ^(x) ^(×n) ^(x) , B₀ ∈ R^(n) ^(x) ^(×n) ^(x) and C_(p) ∈ R^(n) ^(y) ^(×n) ^(x) are known system constant matrices; d_(f)(t) is a disturbance variable; the relative conversion speed n_(h) of the high pressure rotor of the aircraft engine is a scheduling parameter θ∈ R^(p); system variable matrices ΔA(θ) and ΔB(θ) satisfy −ΔA≤ΔA(θ)≤ΔA and −ΔB≤ΔB(θ)≤ΔB; ΔA ∈ R^(n) ^(x) ^(×n) ^(x) is an upper bound of ΔA(θ); ΔB ∈ R^(n) ^(x) ^(×n) ^(u) is an upper bound of ΔB(θ); ΔA≥0, ΔB≥0; a state variable initial value x_(p)(0) satisfies x ₀≤x_(p)(0)≤x ₀; x ₀, x ₀ ∈ R^(n) ^(x) are respectively known upper bound and lower bound of the state variable initial value x_(p)(0); d,d ∈ R^(n) ^(x) are known upper bound and lower bound of an unknown disturbance d_(f)(t); sensor noise v(t) satisfies |v(t)|<V; V is a known bound; V>0; step 1.2: representing the reference model of the fault-free system of the aircraft engine as {dot over (x)}_(pref)(t)=A ₀ x _(pref)(t)+B ₀ u _(pref)(t) y _(pref)(t)=C _(p) x _(pref)(t)   (2) where x_(pref) ∈ R^(n) ^(x) is a reference state vector of the fault-free system; u_(pref) ∈ R^(n) ^(u) is control input of the fault-free system; y_(pref) ∈ R^(n) ^(y) is a reference output vector; an error feedback controller of the fault-free system of the aircraft engine is designed according to the aircraft engine LPV model established in the step 1.1; step 1.2.1: defining an error e_(p)(t)=x_(pref)(t)−x_(p)(t) between the affine parameter-dependent aircraft engine LPV model and the reference model of the fault-free system of the aircraft engine to obtain error state equations of the fault-free system: {dot over (e)}_(p)(t)=[A ₀ +ΔA(θ)]e _(p)(t)+[B ₀ +ΔB(θ)]Δu _(cp)(t)−ΔA(θ)x _(pref)(t)−ΔB(θ)u _(pref)(t)−d _(f)(t) ε_(cp)(t)=C _(p) e _(p)(t)−v(t) where Δu _(cp)(t)=u_(pref)(t)−u_(p)(t) and ε_(cp)(t)=y _(pref)(t)−y _(p)(t); step 1.2.2: representing state equations of the upper bound ē_(p) and the lower bound e _(p) of the error vector e_(p) as: {dot over ( e )}_(p)(t)=[A ₀ −LC _(p) ]ē _(p)(t)+[B ₀ +ΔB]Δu _(cp)(t)+Lε _(cp)(t)+|L|V−d (t)+ΔA |x _(pref)(t)|+ϕ_(p)(t) {dot over ( e )}_(p)(t)=[A ₀ −LC _(p) ]e _(p)(t)+[B ₀ −ΔB]Δu _(cp)(t)+Lε _(cp)(t)−|L|V−d (t)−ΔA |x _(pref)(t)|−ϕ_(p)(t)   (4) where ē_(p), e _(p) ∈ R^(n) ^(x) are respectively the upper bound and the lower bound of the error vector e_(P), i.e., e _(p)(t)≤e_(p)(t)≤ē_(p)(t); ϕ_(p)(t)=ΔA(ē_(p) ⁺(t)+e _(p) ⁻(t)), ē_(p) ⁺=max {0,ē_(p)}, ē_(p) ⁻=ē_(p) ⁺−ē_(p), e _(p) ⁺=max{0,e _(p)}, e _(p) ⁻=e _(p) ⁺−e _(p); L ∈ R^(n) ^(x) ^(×n) ^(y) is an error gain matrix of the fault-free system and satisfies A₀−LC_(p) ∈ M^(n) ^(x) ^(×n) ^(x) ; M^(n) ^(x) represents a set of n_(x)-dimensional Metzler matrix; |L| represents taking absolute values of all elements of the matrix L; step 1.2.3: respectively setting e_(pa)=0.5(ē_(p)+e _(p)) and e_(pd)=ē_(p)−e _(p); rewriting the formula (4) as: {dot over (e)}_(pd)(t)=[A ₀ −LC _(p) ]e _(pd)(t)+2 ΔBΔu _(cp)(t)+ϕ_(pd)(t)+δ_(pd)(t) {dot over (e)}_(pa)(t)=[A ₀ −LC _(p) ]e _(pa)(t)+B ₀ Δu _(cp)(t)+LC _(p) e _(p)(t)+δ_(pa)(t)   (5) where ϕ_(pd)(t)=2 ΔA ( e _(p) ⁺(t)+ e _(p) ⁻(t)) δ_(pd)(t)=2|L|V−d (t)+ d (t)+2 ΔA|x _(pref)(t)|  (6) δ_(pa)(t)=−Lv(t)−0.5( d (t)+ d (t)) step 1.2.4: representing the output of the error feedback controller as: Δu _(cp)(t)=K _(a) e _(pa)(t)+K _(d) e _(pd) (t)   (7) representing the gain matrix of the error feedback controller as K_(d), K_(a) ∈ R^(n) ^(x) ^(×n) ^(x) ; setting e_(x)(t)=e_(p)(t)−e_(pa)(t), −0.5e_(pd)(t)≤e_(x)(t)≤0.5e_(pd)(t), and then {dot over (e)}_(pa)(t)=[A ₀ +B ₀ K _(a)]e _(pa)(t)+B ₀ K _(d) e _(pd)(t)+LC _(p) e _(x)(t)+δ_(pa)(t)   (8) step 1.2.5: rewriting formulas (5) and (8) as: $\begin{matrix} {{{\overset{.}{\xi}}_{p}(t)} = {{{G_{p}(t)}{\xi_{p}(t)}} + {\delta_{p}(t)}}} & (9) \\ {{{{G_{p}(t)} = {\begin{bmatrix} {A_{0} - {LC}_{p}} & 0 \\ {B_{0}K_{d}} & {A_{0} + {B_{0}K_{a}}} \end{bmatrix}\  + {A_{pd}(t)}}}{{where}\mspace{14mu} {\xi_{p}(t)}} = \left\lbrack {{e_{pd}(t)}^{T},{e_{pa}(t)}^{T}} \right\rbrack^{T}},{{\delta_{p}(t)} = {\left\lbrack {\left( {{\delta_{pd}(t)} + {2\overset{\_}{\Delta \; B}\Delta \; {u_{cp}(t)}}} \right)^{T},{\delta_{pa}(t)}^{T}} \right\rbrack^{T}\mspace{14mu} {and}\mspace{14mu} {then}}}} & (10) \\ {\begin{bmatrix} \varphi_{pd} \\ {LC_{p}e_{x}} \end{bmatrix} = {A_{pd}\begin{bmatrix} e_{pd} \\ e_{pa} \end{bmatrix}}} & (11) \end{matrix}$ step 1.2.6: S^(m×m) representing an m-dimensional real symmetric square matrix; setting a matrix E,F ∈ S^(2n) ^(x) ^(×2n) ^(x) ; E,F>0 representing that each element in E,F is greater than 0; constant λ>0; and obtaining a matrix inequality: G _(p) ^(T) E+EG _(p) +λE+F<0   (12) namely, setting each element in G_(p) ^(T)E+EG_(p)+λE+F to be less than 0; solving the matrix inequality (12) to obtain the gain matrices K_(d), K_(a) of the error feedback controller so as to obtain the error feedback controller from (7); step 1.3: describing the aircraft engine LPV model having disturbance and actuator and sensor faults as: {dot over (x)}_(f)(t)=[A ₀ +ΔA(θ)]x _(f)(t)+B _(f)(γ(t))u _(f)(t)+d _(f)(t) y _(f)(t)=C _(f)(ϕ(t))x _(f)(t)+v(t)   (13) where x_(f) ∈ R^(n) ^(x) is a state vector of a system with fault; u_(f) ∈ R^(n) ^(u) is the control input of the system with fault; y_(f) ∈ R^(n) ^(y) is an output vector of the system with fault; B_(f)(γ(t)) and C_(f)(ϕ(t)) are respectively actuator and sensor faults, expressed as B _(f)(γ(t))=[B ₀ +ΔB(θ)]diag(γ₁(t), . . . , γ_(n)(t)) C _(f)(ϕ(t))=C _(p)diag(ϕ₁(t), . . . , ϕ_(n)(t))   (14) where 0≤γ_(i)(t)≤1 and 0≤ϕ_(j)(t)≤1 respectively represent the failure degree of the ith actuator and the jth sensor; γ_(i)=1 and γ₁=0 respectively represent health and complete failure of the ith actuator; ϕ_(j) is similar; diag(γ₁, γ₂, . . . , γ_(n)) represents a diagonal matrix with diagonal elements γ₁, γ₂, . . . , γ_(n); diag(ϕ₁, ϕ₂, . . . , ϕ_(n)) is similar; setting γ(t) and ϕ(t) estimated values respectively as {circumflex over (γ)}(t) and {circumflex over (ϕ)}(t), and then B _(f)(γ(t))=B _(f)({circumflex over (γ)}(t))+B _(f)(Δγ(t)) C _(f)(ϕ(t))=C _(f)({circumflex over (ϕ)}(t))+C _(f)(Δϕ(t))   (15) where Δγ(t)=γ(t)−{circumflex over (γ)}(t) and Δϕ(t)=ϕ(t)−{circumflex over (ϕ)}(t) are respectively errors of estimation of γ(t) and ϕ(t); a virtual actuator and a virtual sensor are respectively designed according to the actuator and sensor faults; step 1.3.1: designing the virtual sensor as: {dot over (x)}_(vs)(t)=A _(vs)(θ)x _(vs)(t)+B _(f)({circumflex over (γ)}(t))Δu(t)+Qy _(f)(t) {circumflex over (y)}_(f)(t)=C _(vs) x _(vs)(t)+Py _(f)(t)   (16) where A _(vs)(θ)=A ₀ +ΔA(θ)−QC _(f)({circumflex over (ϕ)}(t)) C _(vs) =C _(p) −PC _(f)({circumflex over (ϕ)}(t))   (17) where x_(vs) ∈ R^(n) ^(x) is a state variable of a virtual sensor system; Δu ∈ R^(n) ^(u) is a difference in control inputs of a fault model and a fault reference model; ŷ_(f) ∈ R^(n) ^(y) is an output vector of the virtual sensor system; Q and P are respectively parameter matrices of the virtual sensor; step 1.3.2: an LMI region S₁(ρ₁, q₁, r₁, θ₁) representing an intersection of a left half complex plane region with a bound of −ρ₁, a circular region with a radius of r₁ and a circle center of q₁ and a fan region having an intersection angle θ₁ with a negative real axis; representing a state matrix A_(vs) of the virtual sensor as a polytope structure; A_(vsj)=A₀+ΔA(θ_(j))−Q_(j)C_(f)({circumflex over (ϕ)}(t)), where θ_(j) represents the value of the jth vertex θ; A_(vsj) represents the value of the state matrix A_(vs) of the virtual sensor of the jth vertex; a necessary and sufficient condition for eigenvalues of A_(vsj) to be in S₁(ρ₁, q₁, r₁, θ₁) is that there exists a symmetrical matrix X₁>0 so that the linear matrix inequalities (18)-(20) are established, thereby obtaining a parameter matrix Q_(j) of the virtual sensor of the corresponding vertex; $\begin{matrix} {{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} + {X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} + {2\rho_{1}X_{1}}} < 0} & (18) \\ {\begin{bmatrix} {{- r_{1}}X_{1}} & {{q_{1}X_{1}} + {\begin{bmatrix} {A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\ {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}} \end{bmatrix}X_{1}}} \\ {{q_{1}X_{1}} + {X_{1}\begin{bmatrix} {A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\ {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}} \end{bmatrix}}^{T}} & {{- r_{1}}X_{1}} \end{bmatrix} < 0} & (19) \\ {\mspace{20mu} {\left( {\begin{matrix} {\sin \theta_{1}\begin{Bmatrix} {{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} +} \\ {X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} \end{Bmatrix}} \\ {\cos \theta_{1}\begin{Bmatrix} {{X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} -} \\ {\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} \end{Bmatrix}} \end{matrix}\mspace{20mu} \begin{matrix} {\cos \; \theta_{1}\begin{Bmatrix} {{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} -} \\ {X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} \end{Bmatrix}} \\ {\sin \; \theta_{1}\begin{Bmatrix} {{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack X_{1}} +} \\ {X_{1}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {Q_{j}{C_{f}\left( {\hat{\varphi}(t)} \right)}}} \right\rbrack}^{T} \end{Bmatrix}} \end{matrix}} \right) < 0}} & (20) \end{matrix}$ selecting Q_(j) of a vertex corresponding to θ_(j) as a parameter matrix of the virtual sensor; step 1.3.3: representing the parameter matrix P of the virtual sensor as: P=C_(p)C_(f) ^(†)  (21) where † represents pseudo-inversion of the matrix; step 1.3.4: designing the virtual actuator as {dot over (x)}_(va)(t)=A _(va) x _(va)(t)+B _(va) Δu _(c)(t) Δu(t)=Mx _(va)(t)+NΔu _(c)(t)   (22) y _(c)(t)={circumflex over (y)}_(f)(t)+C _(p) x _(va)(t) where A _(va) =A ₀ +ΔA(θ)−B _(f)({circumflex over (γ)}(t))M B _(va) =B ₀ +ΔB(θ)−B _(f)({circumflex over (γ)}(t))N   (23) where x_(va) ∈ R^(n) ^(x) is a state variable of the virtual actuator system; Δu_(c) ∈ R^(n) ^(u) is the output of the error feedback controller; y_(c) ∈ R^(n) ^(y) is an output vector of the virtual actuator system; M and N are respectively parameter matrices of the virtual actuator; step 1.3.5: an LMI region S₂(ρ₂, q₂, r₂, θ₂) representing an intersection of a left half complex plane region with a bound of −ρ₂, a circular region with a radius of r₂ and a circle center of q₂ and a fan region having an intersection angle θ₂ with a negative real axis; representing a state matrix A_(va) of the virtual actuator as a polytope structure; A_(vaj)=A₀+ΔA(θ_(j))−B_(f)({circumflex over (γ)}(t))M_(j), where θ_(j), represents the value of the jth vertex θ; A_(vaj) represents the value of the state matrix A_(va) of the virtual actuator of the jth vertex; a necessary and sufficient condition for eigenvalues of A_(vaj) to be in S₂(ρ₂, q₂, r₂, θ₂) is that there exists a symmetrical matrix X₂>0 so that the linear matrix inequalities (24)-(26) are established, thereby obtaining a parameter matrix M_(i) of the virtual actuator; $\begin{matrix} {{{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} + {X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} + {2\rho_{2}X_{2}}} < 0} & (24) \\ {\begin{bmatrix} {{- r_{2}}X_{2}} & {{q_{2}X_{2}} + {\begin{bmatrix} {A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\ {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}} \end{bmatrix}X_{2}}} \\ {{q_{2}X_{2}} + {X_{2}\begin{bmatrix} {A_{0} + {\Delta A\left( \theta_{j} \right)} -} \\ {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}} \end{bmatrix}}^{T}} & {{- r_{2}}X_{2}} \end{bmatrix} < 0} & (25) \\ {\mspace{20mu} {\left( {\begin{matrix} {\sin \theta_{2}\begin{Bmatrix} {{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} +} \\ {X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} \end{Bmatrix}} \\ {\cos \theta_{2}\begin{Bmatrix} {{X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} -} \\ {\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} \end{Bmatrix}} \end{matrix}\mspace{20mu} \begin{matrix} {\cos \; \theta_{2}\begin{Bmatrix} {{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} -} \\ {X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} \end{Bmatrix}} \\ {\sin \; \theta_{2}\begin{Bmatrix} {{\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack X_{2}} +} \\ {X_{2}\left\lbrack {A_{0} + {\Delta {A\left( \theta_{j} \right)}} - {{B_{f}\left( {\hat{\gamma}(t)} \right)}M_{i}}} \right\rbrack}^{T} \end{Bmatrix}} \end{matrix}} \right) < 0}} & (26) \end{matrix}$ selecting M_(j) of a vertex corresponding to θ_(j) as a parameter matrix of the virtual actuator; step 1.3.6: representing the parameter matrix N of the virtual actuator as: N=B_(f) ^(†)B_(p)   (27) where † represents pseudo-inversion of the matrix; step 1.4: designing an interval error observer according to the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the system with fault; step 1.4.1: representing the reference model of the aircraft engine system having disturbance and actuator and sensor faults as: {dot over (x)}_(ref)(t)=A ₀ x _(ref)(t)+B_(f)({circumflex over (γ)}(t))u _(ref)(t) y _(ref)(t)=C _(f)({circumflex over (ϕ)}(t))x _(ref)(t)   (28) where x_(ref) ∈ R^(n) ^(x) is a reference state vector of the system having disturbance and actuator and sensor faults; u_(ref) ∈ R^(n) ^(u) is control input of the system having disturbance and actuator and sensor faults; y_(ref) ∈ R^(n) ^(y) is a reference output vector of the system having disturbance and actuator and sensor faults; step 1.4.2: defining an error e(t)=x_(ref)(t)−x_(f)(t) between the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the aircraft engine to obtain error state equations of the system with fault of the aircraft engine based on the LPV model: {dot over (e)}(t)=[A ₀ +ΔA(θ)]e(t)+B _(f)({circumflex over (γ)})Δu(t)−B _(f)(Δγ)u _(f)(t)−ΔA(θ)x _(ref)(t)−d _(f)(t) ε_(c)(t)=C _(f)(ϕ(t))e(t)−C _(f)(Δϕ)x _(ref)(t)−v(t)   (29) where Δu(t)=u_(ref)(t)−u_(f)(t) and ε_(c)(t)=y_(ref)(t)−y_(f)(t); step 1.4.3: representing state equations of an upper bound ē and a lower bound e of the error e between the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the aircraft engine as: {dot over ( e )}(t)=[A ₀ −LC _(f)(ϕ(t))] e (t)+[B ₀ +ΔB]Δu _(c)(t)+L[ε _(c)(t)+C _(p) x _(va)+(C _(p) −PC _(f)(ϕ(t)))x _(vs) ]+|L|V−d (t)+ΔA |x _(ref)(t)|+ΔB |u _(ref)|+ϕ(t) {dot over ( e )}(t)=[A ₀ −LC _(f)(ϕ(t))] e (t)+[B ₀ −ΔB]Δu _(c)(t)+L[ε _(c)(t)+C _(p) x _(va)+(C _(p) −PC _(f)(ϕ(t)))x _(vs) ]−|L|V−d (t)−ΔA |x _(ref)(t)|−ΔB |u _(ref)|−ϕ(t)   (30) where ϕ(t)=ΔA(ē_(v) ⁺(t)+e _(v) ⁻(t)), e_(v) is a difference among the error state variable of the system with fault of the aircraft engine based on the LPV model, the state variable of the virtual actuator and the state variable of the virtual sensor; the upper bound of e_(v) is ē_(v)(t)=ē(t)−x_(va)(t)−x_(vs)(t); the lower bound of e_(v) is e _(v)(t)=e(t)−x_(va)(t)−x_(vs)(t); A₀−LC_(f) ∈ M^(n) ^(x) ^(×n) ^(x) ; step 1.4.4: setting e_(a)=0.5(ē+e), e_(d)=ē−e, and obtaining the interval error observer from (30); {dot over (e)}_(d)(t)=[A ₀ −LC _(f)(ϕ(t))]e _(d)(t)+2 ΔBΔu _(c)(t)+ϕ_(d)(t)+δ_(d)(t) {dot over (e)}_(a)(t)=[A ₀ −LC _(f) ]e _(a)(t)+B ₀ K _(a) e _(a)(t)+B ₀ K _(d) e _(d)(t)+δ_(a)(t)+LC _(p) x _(va) +L(C _(p) −PC _(f))+LC _(f) e(t)   (31) where ϕ_(d)(t)=2 ΔA ( e _(v) ⁺(t)+e _(v) ⁻(t)) δ_(d)(t)=2|L|V−d (t)+ d (t)+2 ΔA|x _(ref)(t)|+2 ΔB|u _(ref)(t)|  (32) δ_(a)(t)=−Lv(t)−0.5( d (t)+ d (t)) step 1.5: using the aircraft engine state variable x_(f)(t) of the aircraft engine LPV model having disturbance and actuator and sensor faults, the output variable y_(f)(t), the reference model state variable x_(ref)(t) of the system with fault, the virtual actuator state variable x_(va)(t) and the virtual sensor state variable x_(vs)(t) as inputs of the interval error observer; using the interval error observer output e_(a)(t),e_(d)(t) as the input of the error feedback controller; using the error feedback controller output Δu_(c)(t) as the input of the virtual actuator; inputting the difference between the reference model output u_(ref)(t) of the system with fault and the virtual actuator output Δu(t) as a control signal into the system with fault of the aircraft engine, thereby realizing active fault tolerant control of the aircraft engine. 